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Topic: The Truth About Poincare
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Evelyn Sander

Posts: 187
Registered: 12/3/04
The Truth About Poincare
Posted: May 14, 1993 4:05 PM
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THE TRUTH ABOUT POINCARE

In 1885, King Oscar II of Sweden announced a mathematics contest in
celebration of his sixtieth birthday. The prize would go to the essay
that established the stability of the solar system. It was set up at
the urging of Mittag-Leffler. The judges were quite a distinguished
bunch: Mittag-Leffler, Weierstrass, and Hermite. Thus the winner would
gain great prestige, and his essay would be published in the Swedish
journal Acta Mathematica.

The committee read the many entries from both mathematicians
and astronomers and awarded the prize to Poincare for his essay on
the three body problem. With the hindsight of a hundred years more
research, he deserved the prize; this essay was where he first
developed many of the fundamental ideas which led to the modern field
of Dynamical Systems.

However, though this is how people have told the story for many years,
it is not the whole story. There were actually two essays by Poincare,
only one of which he actually submitted to the contest. Perhaps due to
time constraints, the judges did not read the essays carefully, and it
was only after they awarded the prize to Poincare and published his
original essay that Phragmen pointed out a flaw in Poincare's work.

Poincare had assumed that stable and unstable manifolds do not
intersect transversally. As mentioned in my article on Hamiltonian
Systems, transverse intersections of these manifolds is the key to
much of the interesting behavior. His mistake allowed him to conclude
that he had solved the restricted three body problem. In 1887, he
wrote to Mittag-Leffler: "In this particular case I have found a
rigorous demonstration of stability and a method of placing precise
limits on the elements of the third body."[Goroff]

As soon as Mittag-Leffler heard that there was an error, he wrote to
all of the subscribers to Acta Mathematica recalling the journal. He
then destroyed all but one copy of the original journal; this copy
still remains in a locked drawer in the archives of the Mittag-Leffler
Institute.

The recall of the journal was no secret. The other entrants,
including many Swedish astronomers, now realized that the judges did
not read the essays carefully. The written records of German math
society meetings show quite a bit of debate about the prize scandal.

Rather than picking a new winner in the contest, for the next year,
Mittag-Leffler regularly wrote letters to Poincare asking him when he
would finish the corrected version of the essay. After a year of
this pressure, Poincare came out with a new essay, which
Mittag-Leffler then published and sent to subscribers in the place of
the recalled journal. The new essay did not even claim to have solved
the original problem. However, it was a memorable work in which
Poincare developed the important ideas for which people remember the
contest.

Though people still know the story in Sweden, it had been forgotten in
this country. There was was a reference to it in 1912, when the
scholar F.R.Moulton described the scandal in an article in Popular
Astronomy.[Goroff] More recently, Richard McGehee found out about it
during his stay at the Mittag-Leffler institute. He looked at many of
the old documents from the archives, including copies of the letters
that Mittag-Leffler wrote to Poincare and the one original Acta
Mathematica journal.

I first heard the story in long form from McGehee a few months ago in
his class, and I wish that I could tell it as well. The class was
based around this story. The course announcement described the
contest, saying only that Poincare had won the prize without answering
the stability question and promising to explain what was difficult
about it.

McGehee started the class by a brief description of Poincare's second
essay. We then spent the quarter trying to understand all the
theories on stability and why this is a hard question. The theory
takes you quite far from the original question.

On the last day we were still involved in some number theory that
comes up in the KAM theorem, very far from anything to do with the
solar system. During the break, one classmate mentioned to me that he
thought it was unlikely that he would be able to tie all this in with
the solar system again. It was then, after this three month set-up
which we thought he would not be able to tie together, that McGehee
told us what had really happened with the contest.

Daniel Goroff also heard the story from McGehee. All the references
are to his rendition of the story, published in his introduction to
Poincare's book New Methods of Celestial Mechanics.











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